Abstract

We study the redistribution of Cesium atomic Rydberg states by intense, shaped, narrow-band pulses of millimeter radiation. The radiation source is a large-area photoconductive switch illuminated by a temporally shaped optical pulse. We will present our latest efforts to study atomic redistribution in the strong-field limit using these table-top THz sources.

1. Introduction

Rydberg eigenstates excited by strong radiation fields exhibit non-stationary quantum-dynamics such as wavepacket formation, wavefunction trapping, multiphoton-ionization, and coherent population transfer. The dynamics are particularly rich when the frequency of the driving radiation is near the principal resonance of the system, and the field strength is high enough to have significant probability for a transition during one resonant period. For high Rydberg states with principal quantum number n this period (2πn3 atomic units) is the Kepler period of a classical orbiting electron with the same binding energy, equal to about 2 psec for n=24. Coherent population transfer between neighboring n-states at resonance occurs at the Rabi frequency Ω ∝ Fn2 atomic units where F is the electric field strength, so that the field required to deplete the initial states in one Kepler orbit scales as n-5. Rydberg states with classical orbits of several picoseconds require only a few hundred volts per centimeter.

These dynamics have been studied in various limits, according to the technology which produces the driving field. Many experiments have used nearly-periodic microwave radiation at or near the resonant frequency of Rydberg states with n > 40, where the field strengths required are quite modest. [1–4

Experiments have also utilized broadband coherent field pulses (so-called half-cycle pulses) where the characteristic width of the pulse is close to the classical orbital period, or else is much shorter than the orbital period (impulse limit) [5–7

Finally, pulsed laser experiments have been performed using intense pulses with widths comparable to or shorter than the Kepler period, but optical carrier frequencies much higher than the resonance. Such pulses are commonly used to excite Rydberg wavepackets from ground states or other valence states in atoms or molecules [8

]. The ultimate aim of this research is to discover and explore such structures predicted to occur when the atom is driven strongly on resonance. To fully explore the quantum dynamics, we must go beyond the impulse or periodic limits of previous work to a regime where the shape, spectrum, phase, and polarization of the strong field pulses can be controlled as desired. Means are now available for shaping optical pulses in the far-infrared[19–21

In the experiments reported here, we sculpt 10–150 psec sub-THz pulses by first shaping an optical pulse as follows: Using a standard design Ti:Sapphire Chirped-Pulse Amplified laser system, we produce an optical pulse that has been temporally stretched, shaped using spectral masks, and amplified, but not recompressed. Rapid modulations at frequencies corresponding to energy differences between various Rydberg states are then imposed on this shaped pulse through optical interference in a Michelson interferometer (Figure 1):

When the envelope of the shaped optical pulse matches the desired shape of the THz pulse, the light is directed into a vacuum chamber containing a large-area GaAs photoconductive switch with a large bias in the surface plane [23

]. Care must be taken to restrict the total fluence incident on the GaAs switch to below 40μJ/cm2 to avoid saturation. If this is done, then the frequency, the shape of the pulse train, and the strength of the FIR can all be independently controlled by changing the relative arm length mismatch in the interferometer, by masking the optical stretcher, and by changing the bias field, respectively.

Fig. 2. Pulse shaped to produce a steep rising edge.

Fig. 3. Experimental apparatus.

The shape of the THz field can be monitored by simple cross-correlation of the optical pulses with a recompressed ultrashort optical pulse providing a temporal resolution of less than 1 psec. Figure 2 shows an example of two pulses with nearly the same modulation frequency but different rise times created by spectral shaping in the CPA stretcher. These two pulse shapes were chosen to explore the influence of rapid turn-on of strongly resonant driving fields.

3. Experiment and data

In this experiment, we have studied the excitation of Rydberg states in the n = 20 to 30 range with radiation pulses of variable rise time.

The apparatus is shown in figure 3. The GaAs optical demodulator is housed inside a vacuum chamber in close proximity to a Cs effusive atomic beam. Radiation is collected by an off-axis parabolic metal mirror and directed into the beam. The excitation scheme and the relevant Rydberg states are shown in figure 4.

Atoms in the beam have been prepared in the n=24p state by a multi-step excitation process [22

]: the ground-state 6s atoms are first excited to the 7s state by a two-photon transition using a 10 nsec 1.08μm pulse from an H2 Raman-shifted dye laser pumped by the second harmonic of a Q-switched Nd:YAG laser; then the 7s-state atoms are excited to np Rydberg states by one-photon excitation using a second Nd:YAG-pumped dye laser at around 790 nm.

Following state redistribution by the millimeter-wave pulse, the state of the Rydberg atoms can be analyzed by ramped-field-ionization. A typical ramped electric field signal is shown in figure 5. Several peaks are visible, indicating formation of a Rydberg wavepacket following excitation.

In this experiment we compare population transfer as a function of modulation frequency (i.e. study the line shape) for the two different pulse shapes shown above, as a function of field amplitude. We find that the rising edge of the electric field profile dramatically affects population transfer. Figure 6 shows the excitation probability vs. THz frequency for the case of a slowly rising pulse. Two different curves are displayed, one showing redistribution to states above the 24p state, and the other showing redistribution to states below 24p. The main feature is peaked near the frequency of resonance between the 24p state and the neighboring 24s and 25s states, with additional small structure possibly due to multiphoton resonances to nearby states.

Fig. 4. Grotrian diagram for cesium.

When the pulses are shaped to have a very sharp rise time, the line shapes take on a rich new structure, shown in figure 7. The prominent resonance at around 365 GHz in the redistribution to the 25s state is joined by a satellite of comparable width displaced up by 30–40 GHz. Furthermore, this structure moves by nearly 10 GHz as the electric field amplitude increases by about a factor of 2. This factor is determined by the factor increase in the bias field on the GaAs wafer (see figure 7). Under ideal conditions, the modulated field tracks this bias field. The field in the interaction region may increase less because of saturation in the demodulator. In addition, the estimates of the absolute field in the interaction region are made difficult by the complex shape of the pulses, possible saturation in the demodulator, and the focusing geometry. [24

]. These problems could be overcome in the future by direct electro-optic sampling of the THz field in the atomic beam.

The resonance at 380 GHz in the redistribution down to the 24s state is also joined be satellites when the pulse rising edge is sharpened. These new resonances appear as sharp peaks in a 25 GHz band below the principal resonance. Their behavior with intensity is quite different from the satellite in the up-shifted population. These resonances are sharper, and they shift in amplitude but not in positions as the intensity is raised.

4. Analysis

Specral resonances as the rise time and pulse amplitude are varied will be affected by ac Stark shifts and the accompanying partially adiabatic population transfer among levels close to resonance. Another important effect in our case may be the physics of the transmitter itself. We know from extensive work with half-cycle pulses that the generated electric field is proportional to the external bias applied to the crystal, and hence there is a “unipolar,” or DC component to the modulated field. This can cause DC stark shifts of the energy levels and coupling between the rest of the manifold and the S-state.

Fig. 5a. Field-ionization signal vs. time as a dc field is ramped. The dotted line is the signal recorded when only the 24p state is populated. Redistribution by a millimeter-wave pulse results in additional features (solid line) corresponding to population transfer to neighboring states.

5b. Relevant Rydberg states.

Fig. 6. Redistribution spectra for slowly rising pulse. The top figure show redistribution from 24p up to 25s, and the bottom figure shows redistribution down to 24s.

Fig. 7. Redistribution spectra with shaped pulse with a steep rising edge. The top figure shows redistribution to higher states, while the bottom figure shows redistribution to lower states. The three spectra in each figure correspond to a bias field of 125V/cm (top); 100V/cm (middle); and 75V/cm (bottom).

Fig. 8. Theoretical calculation using slow and steeply rising pulses: Periodic modulation of population with frequency.

We can sort out the explicit role of the pulse shape in these experiments by a direct integration of Schrodinger’s equation for an atom in a time-varying field on a finite basis of bound states. Some results of these calculations are shown in Figure 8. In the low electric field limit (where the Rabi rate is less than the inverse pulse duration) the excitation spectrum is narrow, and the shape is independent of the electric field. However, if the Rabi rate exceeds the inverse pulse duration (as in our case), the excitation probability on line center saturates while at finite detunings it continues to increase, giving rise to a modulated, intensity-dependent spectrum not unlike our data. For the 24p–25s transition and a 150 psec pulse, this can occur for fields greater than 20 V/cm. When combined with a rapidly varying electric field profile, this gives rise to satellite peaks (separated by about the Rabi frequency of 30 GHz) in the excitation spectrum, as seen in Figure 8.

The changes in the excitation spectrum with pulse shape are qualitatively similar to the data. The sideband splitting increases with the pulse amplitude, similar to the satellite observed in the 25s data. Other peaks in the data remain stationary as the pulse amplitude increases. Evidently different mechanisms are responsible for this observed structure, for example DC Stark shifts or multiphoton processes.

5. Conclusion

We have observed resonant population transfer between Rydberg states using tunable, narrow band picosecond electric field pulses. Narrowing the pulse rise time causes sharp satellite structure to emerge in the spectrum of population transfer to neighboring states, with a strong electric field dependence. The pulse peak intensity and residual DC electric field components produce Stark shifts which can cause such satellites.

Following the excitation the final quantum state is a coherent superposition, i.e. a wavepacket. Thus electronic wavepacket motion and recurrences will occur, and might be observed by combining this radiation with ultrafast optical pulse techniques. Such studies open the way for tests of novel quantum-dynamical phenomena in Rydberg wavepackets. Redistribution by these temporally shaped far-infrared pulses can form wavepackets with high angular momentum quantum number l, alter and control wavepacket recurrence spectra and possibly induce dynamical stabilization of Rydberg electron motion.

It is a pleasure to thank Thomas Weinacht for advice and assistance during the preparation of this work. PHB received support from the John Simon Guggenheim Memorial Foundation. This work was supported by the National Science Foundation.

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